find the limit

Question

lim(xe^(1/x)-x)

x->infinity

 

Please help! show all steps

Answer 1

The easiest way is to write your expression in the form f(x) / g(x)  with f(x) = e^1/x -1   and g(x) =  1/x.

 

Then L'Hospital's rule says to replace f/g  with f'/g'    (where prime denotes derivative) and study the result.

 

 

   f'/g'  =   (-1/x²) e^1/x      and g'  = -1/x²   , so  f'/g' =  e^1/x

 

 

  Since exp is a smooth function,   lim_x=>∞ e^1/x   =  exp(lim_x=>∞ 1/x) =  exp(0) = 1